Proper Orthogonal Decomposition for Reduced Order Control of Partial Differential Equations

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Abstract

Numerical models of PDE systems can involve
very large matrix equations, but
feedback controllers for these systems must be computable
in real time to be implemented on physical systems.
Classical control design methods produce
controllers of the same order as the numerical models.
Therefore, emph{reduced} order control design is vital for practical
controllers. The main contribution of this research is a method
of control order reduction
that uses a newly developed low order basis. The low order basis
is obtained
by applying Proper Orthogonal Decomposition (POD) to a set of
functional gains, and is referred to as the functional gain
POD basis.
Low order controllers resulting from the
functional gain POD basis
are compared with low order controllers resulting from
more commonly used time snapshot POD bases, with the two dimensional
heat equation as a test problem. The functional gain
POD basis avoids subjective criteria associated
with the time snapshot POD basis and provides
an equally effective low order controller with larger stability
radii. An efficient and effective methodology is introduced for
using a low order basis in reduced order compensator design. This
method combines
"design-then-reduce" and "reduce-then-design" philosophies.
The desirable qualities of the resulting
reduced order compensator are verified by application
to Burgers' equation in numerical experiments.